Problem 7701
| ODE |
\[ \boxed {t y^{\prime \prime }+\left (-t^{2}+1\right ) y^{\prime }+4 y t=0} \] |
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program solution |
\[ y = c_{1} \left (t^{4}-8 t^{2}+8\right )+c_{2} \left (t^{4}-8 t^{2}+8\right ) \left (\int \frac {{\mathrm e}^{\frac {t^{2}}{2}}}{t \left (t^{4}-8 t^{2}+8\right )^{2}}d t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = c_{1} \left (t^{4}-8 t^{2}+8\right )+c_{2} \left (t^{4}-8 t^{2}+8\right ) \left (\int \frac {{\mathrm e}^{\frac {t^{2}}{2}}}{\left (t^{4}-8 t^{2}+8\right )^{2} t}d t \right ) \] |
Problem 7702
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }-t \left (t +1\right ) y^{\prime }+y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{t} t -c_{2} {\mathrm e}^{t} t \,\operatorname {expIntegral}_{1}\left (t \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{t} c_{1} t +c_{2} {\mathrm e}^{t} t \,\operatorname {expIntegral}_{1}\left (t \right ) \] |
Problem 7703
| ODE |
\[ \boxed {y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y=0} \] |
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program solution |
\[ y = c_{1} \cos \left (2 x \right ) {\mathrm e}^{-x^{2}}+\frac {c_{2} {\mathrm e}^{-x^{2}} \sin \left (2 x \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}} \cos \left (2 x \right )+c_{2} {\mathrm e}^{-x^{2}} \sin \left (2 x \right ) \] |
Problem 7704
| ODE |
\[ \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (z^{2}-1\right )^{\frac {1}{4}} {\left (\sqrt {\lambda +1}\, \left (z +\sqrt {z^{2}-1}\right )\right )}^{\sqrt {\lambda +1}}}{\left (z -1\right )^{\frac {3}{4}} \left (z +1\right )^{\frac {3}{4}}}-\frac {c_{2} \left (z^{2}-1\right )^{\frac {1}{4}} {\left (\sqrt {\lambda +1}\, \left (z +\sqrt {z^{2}-1}\right )\right )}^{-\sqrt {\lambda +1}}}{2 \left (z -1\right )^{\frac {3}{4}} \left (z +1\right )^{\frac {3}{4}} \sqrt {\lambda +1}} \] Verified OK.
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Maple solution | \[ y \left (z \right ) = \frac {c_{1} \left (z +\sqrt {z^{2}-1}\right )^{\sqrt {\lambda +1}}}{\sqrt {z^{2}-1}}+\frac {c_{2} \left (z +\sqrt {z^{2}-1}\right )^{-\sqrt {\lambda +1}}}{\sqrt {z^{2}-1}} \] |
Problem 7705
| ODE |
\[ \boxed {4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{\frac {z}{2}}+c_{2} {\mathrm e}^{\frac {z}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {z}}{2}\right ) \] Verified OK.
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Maple solution | \[ y \left (z \right ) = c_{1} {\mathrm e}^{\frac {z}{2}}+c_{2} {\mathrm e}^{\frac {z}{2}} \left (\int \frac {{\mathrm e}^{-\frac {z}{2}}}{\sqrt {z}}d z \right ) \] |
Problem 7706
| ODE |
\[ \boxed {f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f=0} \] |
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program solution |
\[ f = c_{1} \left (z -1\right ) {\mathrm e}^{-z \left (z -2\right )}+c_{2} \left (-1-i \operatorname {erf}\left (i \left (z -1\right )\right ) \left (z -1\right ) \sqrt {\pi }\, {\mathrm e}^{-\left (z -1\right )^{2}}\right ) \] Verified OK.
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Maple solution | \[ f \left (z \right ) = c_{1} {\mathrm e}^{-z^{2}+2 z} \left (z -1\right )+c_{2} {\mathrm e}^{-z^{2}+2 z} \left (-{\mathrm e}^{-1} \sqrt {\pi }\, \operatorname {erf}\left (i z -i\right ) z +{\mathrm e}^{-1} \sqrt {\pi }\, \operatorname {erf}\left (i z -i\right )+i {\mathrm e}^{z^{2}-2 z}\right ) \] |
Problem 7707
| ODE |
\[ \boxed {z y^{\prime \prime }-2 y^{\prime }+z y=0} \] |
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program solution |
\[ y = c_{1} \left (z -i\right ) {\mathrm e}^{-i z}-\frac {c_{2} \left (i z -1\right ) {\mathrm e}^{i z}}{2} \] Verified OK.
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Maple solution | \[ y \left (z \right ) = c_{1} \left (\cos \left (z \right ) z -\sin \left (z \right )\right )+c_{2} \left (\cos \left (z \right )+\sin \left (z \right ) z \right ) \] |
Problem 7708
| ODE |
\[ \boxed {z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z}=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (2 z -1\right ) z^{2} {\mathrm e}^{-2 z}}{2}-4 c_{2} z^{2} \left (\frac {1}{2}+{\mathrm e}^{-2 z} \operatorname {expIntegral}_{1}\left (-2 z \right ) \left (z -\frac {1}{2}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (z \right ) = c_{1} z^{2} {\mathrm e}^{-2 z} \left (2 z -1\right )+c_{2} z^{2} \left (2 \,\operatorname {expIntegral}_{1}\left (-2 z \right ) z -\operatorname {expIntegral}_{1}\left (-2 z \right )+{\mathrm e}^{2 z}\right ) {\mathrm e}^{-2 z} \] |
Problem 7709
| ODE |
\[ \boxed {y^{\prime \prime }+2 x y^{\prime }+4 y=0} \] |
|
program solution |
\[ y = c_{1} x \,{\mathrm e}^{-x^{2}}+c_{2} \left (\sqrt {\pi }\, \operatorname {erfi}\left (x \right ) x \,{\mathrm e}^{-x^{2}}-1\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} \left (-\sqrt {\pi }\, \operatorname {erfi}\left (x \right ) x +{\mathrm e}^{x^{2}}\right ) \] |
Problem 7710
| ODE |
\[ \boxed {y^{\prime \prime }+x y^{\prime }+3 y=0} \] |
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program solution |
\[ y = c_{1} \left (x^{2}-1\right ) {\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} \left (x^{2}-1\right ) {\mathrm e}^{-\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x^{2}-1\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{2}} \left (x^{2}-1\right )+c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \left (x^{2}-1\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x \right ) \] |
Problem 7711
| ODE |
\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-3 y x=0} \] |
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program solution |
\[ y = c_{1} x \,{\mathrm e}^{\frac {x^{3}}{3}}+c_{2} x \,{\mathrm e}^{\frac {x^{3}}{3}} \left (\int \frac {{\mathrm e}^{-\frac {x^{3}}{3}}}{x^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{3}}{3}} x -\frac {9 c_{2} {\mathrm e}^{\frac {x^{3}}{3}} 3^{\frac {2}{3}} {\mathrm e}^{-\frac {x^{3}}{6}} \left (x^{6} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )+5 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right ) x^{3}+10 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )\right )}{10 x^{3} \left (x^{3}\right )^{\frac {1}{3}}} \] |
Problem 7712
| ODE |
\[ \boxed {\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y=0} \] |
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program solution |
\[ y = \frac {c_{1} x \sqrt {2}}{\left (4 x^{2}-1\right )^{\frac {3}{2}}}-\frac {c_{2} \sqrt {2}\, \left (-2 \ln \left (2 x +\sqrt {4 x^{2}-1}\right ) x +\sqrt {4 x^{2}-1}\right )}{2 \left (4 x^{2}-1\right )^{\frac {3}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x}{\left (4 x^{2}-1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (2 \ln \left (2 x +\sqrt {4 x^{2}-1}\right ) x -\sqrt {4 x^{2}-1}\right )}{\left (4 x^{2}-1\right )^{\frac {3}{2}}} \] |
Problem 7713
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-6 x y^{\prime }+12 y=0} \] |
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program solution |
\[ y = c_{1} \left (x +1\right )^{4}-c_{2} x \left (x^{2}+1\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x^{3}+x \right )+c_{2} \left (x^{4}+6 x^{2}+1\right ) \] |
Problem 7714
| ODE |
\[ \boxed {y^{\prime \prime }+x y^{\prime }+\left (2+x \right ) y=0} \] |
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program solution |
\[ y = c_{1} \left (x^{2}-4 x +3\right ) {\mathrm e}^{-\frac {\left (x -2\right ) x}{2}}+c_{2} \left (x^{2}-4 x +3\right ) {\mathrm e}^{-\frac {\left (x -2\right ) x}{2}} \left (\int \frac {{\mathrm e}^{\frac {x \left (x -4\right )}{2}}}{\left (x^{2}-4 x +3\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {1}{2} x^{2}+x} \left (x^{2}-4 x +3\right )+c_{2} {\mathrm e}^{-\frac {1}{2} x^{2}+x} \left (x^{2}-4 x +3\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{2} x^{2}-2 x}}{\left (x -1\right )^{2} \left (x -3\right )^{2}}d x \right ) \] |
Problem 7715
| ODE |
\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y=0} \] |
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program solution |
\[ y = \frac {c_{1} x 2^{\frac {1}{8}}}{\left (2 x^{2}+1\right )^{\frac {3}{4}}}+\frac {c_{2} x 2^{\frac {7}{8}} \left (\int \frac {1}{\left (2 x^{2}+1\right )^{\frac {1}{4}} x^{2}}d x \right )}{2 \left (2 x^{2}+1\right )^{\frac {3}{4}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x}{\left (2 x^{2}+1\right )^{\frac {3}{4}}}+\frac {c_{2} x \left (\int \frac {1}{\left (2 x^{2}+1\right )^{\frac {1}{4}} x^{2}}d x \right )}{\left (2 x^{2}+1\right )^{\frac {3}{4}}} \] |
Problem 7716
| ODE |
\[ \boxed {4 y^{\prime \prime }+x y^{\prime }+4 y=0} \] |
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program solution |
\[ y = c_{1} x \left (x^{2}-12\right ) {\mathrm e}^{-\frac {x^{2}}{8}}+c_{2} x \left (x^{2}-12\right ) {\mathrm e}^{-\frac {x^{2}}{8}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{8}}}{x^{2} \left (x^{2}-12\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{8}} x \left (x^{2}-12\right )+c_{2} {\mathrm e}^{-\frac {x^{2}}{8}} x \left (x^{2}-12\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{8}}}{\left (x^{2}-12\right )^{2} x^{2}}d x \right ) \] |
Problem 7717
| ODE |
\[ \boxed {y^{\prime \prime }+x y^{\prime }-4 y=0} \] |
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program solution |
\[ y = c_{1} \left (x^{4}+6 x^{2}+3\right )+c_{2} \left (x^{4}+6 x^{2}+3\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{\left (x^{4}+6 x^{2}+3\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x^{4}+6 x^{2}+3\right )+c_{2} \left (x^{4}+6 x^{2}+3\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{\left (x^{4}+6 x^{2}+3\right )^{2}}d x \right ) \] |
Problem 7718
| ODE |
\[ \boxed {4 x y^{\prime \prime }-x y^{\prime }+2 y=0} \] |
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program solution |
\[ y = c_{1} \left (x -8\right ) x +c_{2} \left (\frac {\left (-x^{2}+8 x \right ) \operatorname {expIntegral}_{1}\left (-\frac {x}{4}\right )}{128}-\frac {{\mathrm e}^{\frac {x}{4}} \left (x -4\right )}{32}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x^{2}-8 x \right )+c_{2} \left (\frac {\operatorname {expIntegral}_{1}\left (-\frac {x}{4}\right ) x^{2}}{128}+\frac {{\mathrm e}^{\frac {x}{4}} x}{32}-\frac {\operatorname {expIntegral}_{1}\left (-\frac {x}{4}\right ) x}{16}-\frac {{\mathrm e}^{\frac {x}{4}}}{8}\right ) \] |
Problem 7719
| ODE |
\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (12 x +1\right ) y=0} \] |
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program solution |
\[ y = c_{1} \sqrt {x}\, {\mathrm e}^{-3 x}+c_{2} \sqrt {x}\, {\mathrm e}^{-3 x} \left (\int \frac {{\mathrm e}^{3 x}}{x^{\frac {7}{6}}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {x}\, {\mathrm e}^{-3 x}+c_{2} \sqrt {x}\, {\mathrm e}^{-3 x} \left (\int \frac {{\mathrm e}^{3 x}}{x^{\frac {7}{6}}}d x \right ) \] |
Problem 7720
| ODE |
\[ \boxed {3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y=0} \] |
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program solution |
\[ y = c_{1} \left (x^{2}-2 x +4\right ) {\mathrm e}^{\frac {x}{3}} x^{\frac {2}{3}}+c_{2} \left (x^{2}-2 x +4\right ) {\mathrm e}^{\frac {x}{3}} x^{\frac {2}{3}} \left (\int \frac {x^{\frac {4}{3}} {\mathrm e}^{-\frac {x}{3}}}{\left (x^{2}-2 x +4\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x^{\frac {2}{3}} {\mathrm e}^{\frac {x}{3}} \left (x^{2}-2 x +4\right )+c_{2} x^{\frac {2}{3}} {\mathrm e}^{\frac {x}{3}} \left (x^{2}-2 x +4\right ) \left (\int \frac {x^{\frac {4}{3}} {\mathrm e}^{-\frac {x}{3}}}{\left (x^{2}-2 x +4\right )^{2}}d x \right ) \] |
Problem 7721
| ODE |
\[ \boxed {2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y=0} \] |
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program solution |
\[ y = c_{1} \left (x^{4}-9 x^{3}+\frac {63}{4} x^{2}\right )+16 c_{2} \left (x^{2}-9 x +\frac {63}{4}\right ) x^{2} \left (\int \frac {{\mathrm e}^{x}}{x^{\frac {7}{2}} \left (4 x^{2}-36 x +63\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x^{2} \left (4 x^{2}-36 x +63\right )+c_{2} x^{2} \left (4 x^{2}-36 x +63\right ) \left (\int \frac {{\mathrm e}^{x}}{\left (4 x^{2}-36 x +63\right )^{2} x^{\frac {7}{2}}}d x \right ) \] |
Problem 7722
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (2 x +1\right ) y=0} \] |
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program solution |
\[ y = c_{1} \sqrt {x}-c_{2} \sqrt {x}\, \operatorname {expIntegral}_{1}\left (-x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {x}+c_{2} \sqrt {x}\, \operatorname {expIntegral}_{1}\left (-x \right ) \] |
Problem 7723
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (1-2 x \right ) y=0} \] |
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program solution |
\[ y = \frac {c_{1}}{x}-\frac {c_{2} \operatorname {expIntegral}_{1}\left (-2 x \right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} \operatorname {expIntegral}_{1}\left (-2 x \right )}{x} \] |
Problem 7724
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (-x +4\right ) y=0} \] |
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program solution |
\[ y = c_{1} \left (x^{2}+4 x +2\right ) x^{2} {\mathrm e}^{x}-\frac {c_{2} x^{2} \left (\left (x^{2}+4 x +2\right ) {\mathrm e}^{x} \operatorname {expIntegral}_{1}\left (x \right )-x -3\right )}{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x} x^{2} \left (x^{2}+4 x +2\right )-\frac {c_{2} x^{2} \left (-x^{2} \operatorname {expIntegral}_{1}\left (x \right )+{\mathrm e}^{-x} x -4 \,\operatorname {expIntegral}_{1}\left (x \right ) x +3 \,{\mathrm e}^{-x}-2 \,\operatorname {expIntegral}_{1}\left (x \right )\right ) {\mathrm e}^{x}}{4} \] |
Problem 7725
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (3-x \right ) y^{\prime }+y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (x -1\right )}{x}+\frac {c_{2} \left (-\operatorname {expIntegral}_{1}\left (-x \right ) x +\operatorname {expIntegral}_{1}\left (-x \right )-{\mathrm e}^{x}\right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x -1\right )}{x}+\frac {c_{2} \left (\operatorname {expIntegral}_{1}\left (-x \right ) x -\operatorname {expIntegral}_{1}\left (-x \right )+{\mathrm e}^{x}\right )}{x} \] |
Problem 7726
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\sqrt {3}\, x} x^{\sqrt {5}-\frac {1}{2}}+\frac {c_{2} \sqrt {3}\, x^{\sqrt {5}-\frac {1}{2}} {\mathrm e}^{\sqrt {3}\, x}}{6} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{\sqrt {5}} \sinh \left (\sqrt {3}\, x \right )}{\sqrt {x}}+\frac {c_{2} x^{\sqrt {5}} \cosh \left (\sqrt {3}\, x \right )}{\sqrt {x}} \] |
Problem 7727
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (-x +4\right ) y=0} \] |
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program solution |
\[ y = c_{1} x^{2} {\mathrm e}^{-x}-c_{2} x^{2} {\mathrm e}^{-x} \operatorname {expIntegral}_{1}\left (-x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} x^{2}+c_{2} x^{2} {\mathrm e}^{-x} \operatorname {expIntegral}_{1}\left (-x \right ) \] |
Problem 7728
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (2+x \right ) y=0} \] |
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program solution |
\[ y = \frac {c_{1} {\mathrm e}^{-x}}{x}+\frac {c_{2} \left (x^{2}-2 x +2\right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}-2 x +2\right )}{x}+\frac {c_{2} {\mathrm e}^{-x}}{x} \] |
Problem 7729
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y=0} \] |
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program solution |
\[ y = \frac {c_{1}}{\sqrt {x}}-\frac {c_{2} \left (2 x +1\right ) {\mathrm e}^{-2 x}}{4 \sqrt {x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{-2 x} \left (2 x +1\right )}{\sqrt {x}} \] |
Problem 7730
| ODE |
\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (2+x \right )}{x}+\frac {c_{2} \left (4+\ln \left (x +1\right ) \left (2+x \right )\right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x +2\right )}{x}+\frac {c_{2} \left (\ln \left (x +1\right ) x +2 \ln \left (x +1\right )+4\right )}{x} \] |
Problem 7731
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (x^{2}+3\right )}{x^{2}}+\frac {c_{2} \left (x^{2}+3\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{x^{2} \left (x^{2}+3\right )^{2}}d x \right )}{x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+3\right )}{x^{2}}+\frac {c_{2} \left (x^{2}+3\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{x^{2} \left (x^{2}+3\right )^{2}}d x \right )}{x^{2}} \] |
Problem 7732
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y=0} \] |
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program solution |
\[ y = \frac {c_{1} {\mathrm e}^{x}}{x}+\frac {c_{2} \left (-x -1\right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x +1\right )}{x}+\frac {c_{2} {\mathrm e}^{x}}{x} \] |
Problem 7733
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y=0} \] |
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program solution |
\[ y = c_{1} \left (x +1\right ) x^{2} {\mathrm e}^{x}-c_{2} x^{2} \left (-1+{\mathrm e}^{x} \left (x +1\right ) \operatorname {expIntegral}_{1}\left (x \right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x} x^{2} \left (x +1\right )+c_{2} x^{2} \left (-\operatorname {expIntegral}_{1}\left (x \right ) x -\operatorname {expIntegral}_{1}\left (x \right )+{\mathrm e}^{-x}\right ) {\mathrm e}^{x} \] |
Problem 7734
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (2+x \right )}{x}+\frac {c_{2} \left (x -2\right ) {\mathrm e}^{x}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x +2\right )}{x}+\frac {c_{2} {\mathrm e}^{x} \left (x -2\right )}{x} \] |
Problem 7735
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\right ) y=0} \] |
|
program solution |
\[ y = c_{1} \left (4+x \right ) {\mathrm e}^{x} x^{2}+\frac {c_{2} \left ({\mathrm e}^{x} x^{3} \left (4+x \right ) \operatorname {expIntegral}_{1}\left (x \right )-x^{3}-3 x^{2}+2 x -2\right )}{24 x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} x^{2} {\mathrm e}^{x} \left (x +4\right )-\frac {c_{2} \left (-\operatorname {expIntegral}_{1}\left (x \right ) x^{4}+{\mathrm e}^{-x} x^{3}-4 \,\operatorname {expIntegral}_{1}\left (x \right ) x^{3}+3 x^{2} {\mathrm e}^{-x}-2 \,{\mathrm e}^{-x} x +2 \,{\mathrm e}^{-x}\right ) {\mathrm e}^{x}}{24 x} \] |
Problem 7736
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x^{2}-4 x +2\right )}{x^{2}}+\frac {c_{2} \left (\left (-x^{2}+4 x -2\right ) \operatorname {expIntegral}_{1}\left (-x \right )-{\mathrm e}^{x} \left (x -3\right )\right )}{4 x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}-4 x +2\right )}{x^{2}}+\frac {c_{2} \left (\frac {x^{2} \operatorname {expIntegral}_{1}\left (-x \right )}{4}+\frac {x \,{\mathrm e}^{x}}{4}-\operatorname {expIntegral}_{1}\left (-x \right ) x -\frac {3 \,{\mathrm e}^{x}}{4}+\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{2}\right )}{x^{2}} \] |
Problem 7737
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} {\mathrm e}^{x}}{x^{\frac {3}{2}}}-\frac {c_{2} \left (x^{2}+2 x +2\right )}{x^{\frac {3}{2}}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+2 x +2\right )}{x^{\frac {3}{2}}}+\frac {c_{2} {\mathrm e}^{x}}{x^{\frac {3}{2}}} \] |
Problem 7738
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (x +1\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1}}{x}+\frac {c_{2} \left (2 x \,\operatorname {expIntegral}_{1}\left (2 x \right )-{\mathrm e}^{-2 x}\right )}{x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} \left (2 \,\operatorname {expIntegral}_{1}\left (2 x \right ) x -{\mathrm e}^{-2 x}\right )}{x^{2}} \] |
Problem 7739
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y=0} \] |
|
program solution |
\[ y = c_{1} x -c_{2} \operatorname {expIntegral}_{1}\left (x \right ) x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} x +\operatorname {expIntegral}_{1}\left (x \right ) x c_{2} \] |
Problem 7740
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} {\mathrm e}^{-2 x}}{\sqrt {x}}+\frac {c_{2}}{2 \sqrt {x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{-2 x}}{\sqrt {x}} \] |
Problem 7741
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (4+x \right ) y^{\prime }+\left (2+x \right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1}}{x}+\frac {c_{2} \left (\operatorname {expIntegral}_{1}\left (x \right ) x -{\mathrm e}^{-x}\right )}{x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} \left (-\operatorname {expIntegral}_{1}\left (x \right ) x +{\mathrm e}^{-x}\right )}{x^{2}} \] |
Problem 7742
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x -i\right ) {\mathrm e}^{-i x}}{x^{\frac {3}{2}}}-\frac {c_{2} \left (i x -1\right ) {\mathrm e}^{i x}}{2 x^{\frac {3}{2}}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{i x} \left (x +i\right )}{x^{\frac {3}{2}}}+\frac {c_{2} {\mathrm e}^{-i x} \left (x -i\right )}{x^{\frac {3}{2}}} \] |
Problem 7743
| ODE |
\[ \boxed {x y^{\prime \prime }+2 y^{\prime }+y x=0} \] |
|
program solution |
\[ y = \frac {c_{1} \cos \left (x \right )}{x}+\frac {c_{2} \sin \left (x \right )}{x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )}{x}+\frac {c_{2} \cos \left (x \right )}{x} \] |
Problem 7744
| ODE |
\[ \boxed {2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (10 x +1\right )}{10 x^{\frac {3}{2}}}+\frac {c_{2} \left (100 x +10\right ) \left (\int \frac {\sqrt {x}\, {\mathrm e}^{5 x}}{\left (10 x +1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (10 x +1\right )}{x^{\frac {3}{2}}}+\frac {c_{2} \left (10 x +1\right ) \left (\int \frac {\sqrt {x}\, {\mathrm e}^{5 x}}{\left (10 x +1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \] |
Problem 7745
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \cos \left (x \right )}{\sqrt {x}}+\frac {c_{2} \sin \left (x \right )}{\sqrt {x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )}{\sqrt {x}}+\frac {c_{2} \cos \left (x \right )}{\sqrt {x}} \] |
Problem 7746
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y=0} \] |
|
program solution |
\[ y = c_{1} \left (x -n \right ) {\mathrm e}^{-x}-c_{2} \left (n -x \right ) {\mathrm e}^{-x} \left (\int \frac {x^{-n} {\mathrm e}^{x}}{\left (n -x \right )^{2}}d x \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-x} \left (-x +n \right )+c_{2} {\mathrm e}^{-x} \left (-x +n \right ) \left (\int \frac {{\mathrm e}^{x} x^{-n}}{\left (-x +n \right )^{2}}d x \right ) \] |
Problem 7747
| ODE |
\[ \boxed {x^{4} y^{\prime \prime }+x y^{\prime }+y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x^{2}-1\right )}{x}+\frac {c_{2} \left (x^{2}-1\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{2 x^{2}}} x^{2}}{\left (x^{2}-1\right )^{2}}d x \right )}{x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}-1\right )}{x}+\frac {c_{2} \left (x^{2}-1\right ) \left (\int \frac {x^{2} {\mathrm e}^{\frac {1}{2 x^{2}}}}{\left (x +1\right )^{2} \left (x -1\right )^{2}}d x \right )}{x} \] |
Problem 7748
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (2 x +3\right ) {\mathrm e}^{-2 x}}{2 x^{2}}+\frac {c_{2} \left (2 x^{2}-4 x +3\right )}{4 x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (2 x^{2}-4 x +3\right )}{x^{2}}+\frac {c_{2} {\mathrm e}^{-2 x} \left (2 x +3\right )}{2 x^{2}} \] |
Problem 7749
| ODE |
\[ \boxed {\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y=0} \] |
|
program solution |
\[ y = c_{1} \left (x -1\right )+c_{2} \left (2+4 x \right ) \sqrt {x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} \left (x -1\right )+c_{2} \sqrt {x}\, \left (2 x +1\right ) \] |
Problem 7750
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1}}{x}-\frac {c_{2} {\mathrm e}^{-x} \left (x^{2}+2 x +2\right )}{x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} {\mathrm e}^{-x} \left (x^{2}+2 x +2\right )}{x} \] |
Problem 7751
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-x^{2} y^{\prime }+\left (x -2\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} {\mathrm e}^{x}}{x}+\frac {c_{2} \left (-x^{2}-2 x -2\right )}{x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+2 x +2\right )}{x}+\frac {c_{2} {\mathrm e}^{x}}{x} \] |
Problem 7752
| ODE |
\[ \boxed {x^{2} \left (1-4 x \right ) y^{\prime \prime }-\frac {x y^{\prime }}{2}-\frac {3 y x}{4}=0} \] |
|
program solution |
\[ y = \frac {c_{1} x^{\frac {1}{4}} \sqrt {2}\, \left (\sqrt {1-4 x}+1\right ) \sqrt {\frac {\sqrt {1-4 x}+1}{\sqrt {x}}}}{4}+\frac {4 c_{2} x^{\frac {7}{4}} \sqrt {2}\, \sqrt {\frac {\sqrt {1-4 x}+1}{\sqrt {x}}}\, \sqrt {1-4 x}}{3 \left (\sqrt {1-4 x}+1\right )^{2} \sqrt {4 x -1}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {3 x -1}\, \left (\frac {4 \sqrt {\left (-1+4 x \right ) x}\, x +8 x^{2}-2 \sqrt {\left (-1+4 x \right ) x}-5 x +1}{5 x -1+2 \sqrt {\left (-1+4 x \right ) x}}\right )^{\frac {1}{4}}+c_{2} \sqrt {3 x -1}\, \left (\frac {5 x -1+2 \sqrt {\left (-1+4 x \right ) x}}{4 \sqrt {\left (-1+4 x \right ) x}\, x +8 x^{2}-2 \sqrt {\left (-1+4 x \right ) x}-5 x +1}\right )^{\frac {1}{4}} \] |
Problem 7753
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (-9+x \right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x^{2}-8 x +20\right )}{x^{3}}-\frac {c_{2} \left (x^{3}+9 x^{2}+36 x +60\right ) {\mathrm e}^{-x}}{x^{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}-8 x +20\right )}{x^{3}}+\frac {c_{2} {\mathrm e}^{-x} \left (x^{3}+9 x^{2}+36 x +60\right )}{x^{3}} \] |
Problem 7754
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y=0} \] |
|
program solution |
\[ y = c_{1} \left (x -3\right ) x \,{\mathrm e}^{-x}+\frac {c_{2} \left (-{\mathrm e}^{-x} x^{2} \left (x -3\right ) \operatorname {expIntegral}_{1}\left (-x \right )-x^{2}+2 x +1\right )}{6 x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-x} \left (x -3\right )+\frac {c_{2} \left (\operatorname {expIntegral}_{1}\left (-x \right ) x^{3}+{\mathrm e}^{x} x^{2}-3 x^{2} \operatorname {expIntegral}_{1}\left (-x \right )-2 x \,{\mathrm e}^{x}-{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{6 x} \] |
Problem 7755
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y=0} \] |
|
program solution |
\[ y = c_{1} x^{4} {\mathrm e}^{x}+\frac {c_{2} x \left (\operatorname {expIntegral}_{1}\left (x \right ) x^{3} {\mathrm e}^{x}-x^{2}+x -2\right )}{6} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x} x^{4}-\frac {c_{2} x \,{\mathrm e}^{x} \left (-\operatorname {expIntegral}_{1}\left (x \right ) x^{3}+x^{2} {\mathrm e}^{-x}-{\mathrm e}^{-x} x +2 \,{\mathrm e}^{-x}\right )}{6} \] |
Problem 7756
| ODE |
\[ \boxed {2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (5 x +2\right )}{5 \sqrt {x}}+\frac {c_{2} \left (25 x +10\right ) \left (\int \frac {x^{\frac {5}{2}} {\mathrm e}^{-\frac {1}{x}}}{\left (5 x +2\right )^{2}}d x \right )}{\sqrt {x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (5 x +2\right )}{\sqrt {x}}+\frac {c_{2} \left (5 x +2\right ) \left (\int \frac {x^{\frac {5}{2}} {\mathrm e}^{-\frac {1}{x}}}{\left (5 x +2\right )^{2}}d x \right )}{\sqrt {x}} \] |
Problem 7757
| ODE |
\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x \left (x -1\right )\right )^{\frac {1}{4}}}{\left (x -1\right )^{\frac {1}{4}} x^{\frac {3}{4}}}+\frac {c_{2} \left (x \left (x -1\right )\right )^{\frac {1}{4}} \left (-\ln \left (2\right )+\ln \left (2 x -1+2 \sqrt {x \left (x -1\right )}\right )\right )}{\left (x -1\right )^{\frac {1}{4}} x^{\frac {3}{4}}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} \ln \left (x -\frac {1}{2}+\sqrt {x \left (x -1\right )}\right )}{\sqrt {x}} \] |
Problem 7758
| ODE |
\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = c_{1} x +c_{2} \left (\arctan \left (\sqrt {x -1}\right ) x -\sqrt {x -1}\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} \left (\arctan \left (\sqrt {x -1}\right ) x -\sqrt {x -1}\right ) \] |
Problem 7759
| ODE |
\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x +1\right ) \sqrt {x}}{\left (x -1\right )^{4}}+\frac {c_{2} \left (2 x^{2}+12 x +2\right )}{\left (x -1\right )^{4}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+6 x +1\right )}{\left (x -1\right )^{4}}+\frac {c_{2} \sqrt {x}\, \left (x +1\right )}{\left (x -1\right )^{4}} \] |
Problem 7760
| ODE |
\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (6 x -1\right ) x^{\frac {1}{6}} \left (x -1\right )^{\frac {5}{6}}}{6 \left (x \left (x -1\right )\right )^{\frac {1}{6}}}-\frac {9 c_{2} \left (6 x -5\right ) \left (x -1\right )^{\frac {1}{6}} x^{\frac {5}{6}}}{5 \left (x \left (x -1\right )\right )^{\frac {1}{6}}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} \left (6 x -5\right ) x^{\frac {2}{3}}+c_{2} \left (6 x -1\right ) \left (x -1\right )^{\frac {2}{3}} \] |
Problem 7761
| ODE |
\[ \boxed {4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}}=0} \] |
|
program solution |
\[ y = c_{1} \left (x -1\right )^{\frac {3}{4}} \left (x +1\right )^{\frac {3}{4}}-c_{2} \left (x -1\right )^{\frac {1}{4}} \left (x +1\right )^{\frac {1}{4}} x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{\frac {1}{4}} x +c_{2} \left (x^{2}-1\right )^{\frac {3}{4}} \] |
Problem 7762
| ODE |
\[ \boxed {u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u=0} \] |
|
program solution |
\[ u = \frac {c_{1} \left (a x +1\right ) {\mathrm e}^{-a x}}{a}+\frac {c_{2} \left (a x -1\right ) {\mathrm e}^{a x}}{2 a^{2}} \] Verified OK.
|
|
Maple solution | \[ u \left (x \right ) = c_{1} {\mathrm e}^{a x} \left (a x -1\right )+\frac {c_{2} {\mathrm e}^{-a x} \left (a x +1\right )}{a} \] |
Problem 7763
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u=0} \] |
|
program solution |
\[ u = \frac {c_{1} {\mathrm e}^{a x}}{x}-\frac {c_{2} {\mathrm e}^{-a x}}{2 a x} \] Verified OK.
|
|
Maple solution | \[ u \left (x \right ) = \frac {c_{1} \sinh \left (a x \right )}{x}+\frac {c_{2} \cosh \left (a x \right )}{x} \] |
Problem 7764
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u=0} \] |
|
program solution |
\[ u = \frac {c_{1} {\mathrm e}^{\sqrt {-a^{2}}\, x}}{x}+\frac {c_{2} \sqrt {-a^{2}}\, {\mathrm e}^{-\sqrt {-a^{2}}\, x}}{2 a^{2} x} \] Verified OK.
|
|
Maple solution | \[ u \left (x \right ) = \frac {c_{1} \sin \left (a x \right )}{x}+\frac {c_{2} \cos \left (a x \right )}{x} \] |
Problem 7765
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u=0} \] |
|
program solution |
\[ u = \frac {c_{1} \left (a x +1\right ) {\mathrm e}^{-a x}}{a \,x^{3}}+\frac {c_{2} \left (a x -1\right ) {\mathrm e}^{a x}}{2 a^{2} x^{3}} \] Verified OK.
|
|
Maple solution | \[ u \left (x \right ) = \frac {c_{1} {\mathrm e}^{a x} \left (a x -1\right )}{x^{3}}+\frac {c_{2} {\mathrm e}^{-a x} \left (a x +1\right )}{x^{3} a} \] |
Problem 7766
| ODE |
\[ \boxed {u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u=0} \] |
|
program solution |
\[ u = \frac {c_{1} \left (a x -i\right ) {\mathrm e}^{-i a x}}{x^{3} a}-\frac {c_{2} \left (i a x -1\right ) {\mathrm e}^{i a x}}{2 a^{2} x^{3}} \] Verified OK.
|
|
Maple solution | \[ u \left (x \right ) = \frac {c_{1} \left (\cos \left (a x \right ) a x -\sin \left (a x \right )\right )}{x^{3}}+\frac {c_{2} \left (\cos \left (a x \right )+\sin \left (a x \right ) a x \right )}{x^{3}} \] |
Problem 7767
| ODE |
\[ \boxed {y^{\prime \prime }-a^{2} y-\frac {6 y}{x^{2}}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (a^{2} x^{2}+3 a x +3\right ) {\mathrm e}^{-a x}}{a^{2} x^{2}}+\frac {c_{2} {\mathrm e}^{a x} \left (a^{2} x^{2}-3 a x +3\right )}{2 a^{3} x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-a x} \left (a^{2} x^{2}+3 a x +3\right )}{x^{2} a^{2}}+\frac {c_{2} {\mathrm e}^{a x} \left (a^{2} x^{2}-3 a x +3\right )}{3 x^{2}} \] |
Problem 7768
| ODE |
\[ \boxed {y^{\prime \prime }+n^{2} y-\frac {6 y}{x^{2}}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (n^{2} x^{2}-3 i n x -3\right ) {\mathrm e}^{-i n x}}{x^{2} n^{2}}-\frac {c_{2} {\mathrm e}^{i n x} \left (i n^{2} x^{2}-3 x n -3 i\right )}{2 n^{3} x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (\cos \left (n x \right ) x^{2} n^{2}-3 \sin \left (n x \right ) n x -3 \cos \left (n x \right )\right )}{x^{2}}+\frac {c_{2} \left (\sin \left (n x \right ) x^{2} n^{2}+3 \cos \left (n x \right ) n x -3 \sin \left (n x \right )\right )}{x^{2}} \] |
Problem 7769
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\frac {1}{4}\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} {\mathrm e}^{-x}}{\sqrt {x}}+\frac {c_{2} {\mathrm e}^{x}}{2 \sqrt {x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {c_{1} \sinh \left (x \right )}{\sqrt {x}}+\frac {c_{2} \cosh \left (x \right )}{\sqrt {x}} \] |
Problem 7770
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (-i a +x \right ) {\mathrm e}^{-\frac {i x}{a}}}{x^{\frac {3}{2}}}-\frac {c_{2} \left (i x -a \right ) a \,{\mathrm e}^{\frac {i x}{a}}}{2 x^{\frac {3}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {i x}{a}} \left (-i x +a \right )}{x^{\frac {3}{2}}}+\frac {c_{2} {\mathrm e}^{-\frac {i x}{a}} \left (i x +a \right )}{x^{\frac {3}{2}}} \] |
Problem 7771
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x^{2}-3 i x -3\right ) {\mathrm e}^{-i x}}{x^{\frac {5}{2}}}-\frac {c_{2} {\mathrm e}^{i x} \left (i x^{2}-3 x -3 i\right )}{2 x^{\frac {5}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{i x} \left (x^{2}+3 i x -3\right )}{x^{\frac {5}{2}}}+\frac {c_{2} {\mathrm e}^{-i x} \left (x^{2}-3 i x -3\right )}{x^{\frac {5}{2}}} \] |
Problem 7772
| ODE |
\[ \boxed {y^{\prime \prime }+q y^{\prime }-\frac {2 y}{x^{2}}=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (q x +2\right ) {\mathrm e}^{-q x}}{q x}+\frac {c_{2} \left (q x -2\right )}{q^{2} x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (q x -2\right )}{x}+\frac {c_{2} {\mathrm e}^{-q x} \left (q x +2\right )}{q x} \] |
Problem 7773
| ODE |
\[ \boxed {x y^{\prime \prime }+3 y^{\prime }+4 x^{3} y=0} \] |
|
program solution |
\[ y = \frac {c_{1} {\mathrm e}^{-i x^{2}}}{x^{2}}-\frac {i c_{2} {\mathrm e}^{i x^{2}}}{4 x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \sin \left (x^{2}\right )}{x^{2}}+\frac {c_{2} \cos \left (x^{2}\right )}{x^{2}} \] |
Problem 7774
| ODE |
\[ \boxed {\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} x^{\frac {3}{2}} \sqrt {x \left (2+x \right )}}{\sqrt {2+x}}+\frac {c_{2} \sqrt {x \left (2+x \right )}\, \left (-x -1\right )}{\sqrt {x}\, \sqrt {2+x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x +1\right )+c_{2} x^{2} \] |
Problem 7775
| ODE |
\[ \boxed {\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} x^{\frac {3}{2}} \sqrt {x \left (2+x \right )}}{\sqrt {2+x}}+\frac {c_{2} \sqrt {x \left (2+x \right )}\, \left (-x -1\right )}{\sqrt {x}\, \sqrt {2+x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x +1\right )+c_{2} x^{2} \] |
Problem 7776
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x^{2}+1\right )^{2}}{\left (i x +1\right )^{2}}+\frac {c_{2} \left (x^{2}+1\right )^{2} x}{\left (-x +i\right )^{2} \left (x +i\right )^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} \left (x^{2}-1\right ) \] |
Problem 7777
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x^{2}+1\right )^{2}}{\left (i x +1\right )^{2}}+\frac {c_{2} \left (x^{2}+1\right )^{2} x}{\left (-x +i\right )^{2} \left (x +i\right )^{2}} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} \left (x^{2}-1\right ) \] |
Problem 7778
| ODE |
\[ \boxed {y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{x^{2}}+c_{2} x \,{\mathrm e}^{x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x^{2}}+c_{2} x \,{\mathrm e}^{x^{2}} \] |
Problem 7779
| ODE |
\[ \boxed {y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-2\right ) y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{x^{2}}+c_{2} x \,{\mathrm e}^{x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x^{2}}+c_{2} x \,{\mathrm e}^{x^{2}} \] |
Problem 7780
| ODE |
\[ \boxed {\left (2 x -3\right ) y^{\prime \prime }-x y^{\prime }+y=0} \] |
|
program solution |
\[ y = c_{1} 2^{\frac {3}{8}} x +\frac {c_{2} 2^{\frac {5}{8}} x \left (\int \frac {{\mathrm e}^{\frac {x}{2}} \left (2 x -3\right )^{\frac {3}{4}}}{x^{2}}d x \right )}{2} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} x \left (\int \frac {\left (-3+2 x \right )^{\frac {3}{4}} {\mathrm e}^{\frac {x}{2}}}{x^{2}}d x \right ) \] |
Problem 7781
| ODE |
\[ \boxed {y^{\prime \prime }-x y^{\prime }-3 y=0} \] |
|
program solution |
\[ y = c_{1} \left (x^{2}+1\right ) {\mathrm e}^{\frac {x^{2}}{2}}+c_{2} \left (x^{2}+1\right ) {\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{\left (x^{2}+1\right )^{2}}d x \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \left (x^{2}+1\right )+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \left (x^{2}+1\right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{\left (x^{2}+1\right )^{2}}d x \right ) \] |
Problem 7782
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+y=0} \] |
|
program solution |
\[ y = c_{1} x +c_{2} \left (\operatorname {arcsinh}\left (x \right ) x -\sqrt {x^{2}+1}\right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} \left (\operatorname {arcsinh}\left (x \right ) x -\sqrt {x^{2}+1}\right ) \] |
Problem 7783
| ODE |
\[ \boxed {y^{\prime \prime }-x y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = c_{1} \left (x^{2}-1\right )+c_{2} \left (x^{2}-1\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x^{2}-1\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )+c_{2} \left (x^{2}-1\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x \right ) \] |
Problem 7784
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime }+y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x +\sqrt {x^{2}-1}\right )^{\frac {\sqrt {5}}{2}} \sqrt {2 x +3}\, \left (5 x -5\right )^{\frac {1}{4}} \sqrt {\frac {x +1}{\sqrt {-x^{2}+1}}}}{\sqrt {\frac {i \left (3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}\right )}{2 x +3}}\, \left (x +1\right )^{\frac {1}{4}}}+\frac {i c_{2} \left (x +\sqrt {x^{2}-1}\right )^{\frac {\sqrt {5}}{2}} \sqrt {2 x +3}\, \left (5 x -5\right )^{\frac {1}{4}} \sqrt {\frac {x +1}{\sqrt {-x^{2}+1}}}\, \left (\int \frac {\sqrt {x +1}\, \left (3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}\right ) \left (x +\sqrt {x^{2}-1}\right )^{-\sqrt {5}}}{\left (2 x +3\right )^{2} \sqrt {5 x -5}}d x \right )}{\sqrt {\frac {i \left (3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}\right )}{2 x +3}}\, \left (x +1\right )^{\frac {1}{4}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {2 x +3}\, {\left (\frac {3 \sqrt {5}\, x +2 \sqrt {5}-5 \sqrt {x^{2}-1}}{3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}}\right )}^{\frac {1}{4}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {3 \sqrt {5}}{10}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {\sqrt {5}}{5}}+c_{2} \sqrt {2 x +3}\, {\left (\frac {3 \sqrt {5}\, x +2 \sqrt {5}+5 \sqrt {x^{2}-1}}{3 \sqrt {5}\, x +2 \sqrt {5}-5 \sqrt {x^{2}-1}}\right )}^{\frac {1}{4}} \left (x +\sqrt {x^{2}-1}\right )^{-\frac {3 \sqrt {5}}{10}} \left (x +\sqrt {x^{2}-1}\right )^{-\frac {\sqrt {5}}{5}} \] |
Problem 7785
| ODE |
\[ \boxed {x \left (x +1\right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+\left (x -1\right ) y=0} \] |
|
program solution |
\[ y = c_{1} \left (x +1\right )+c_{2} \left (x +1\right ) \ln \left (x \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} \left (x +1\right )+c_{2} \left (x +1\right ) \ln \left (x \right ) \] |
Problem 7786
| ODE |
\[ \boxed {2 x y^{\prime \prime }-y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (2 \sqrt {-x}-1\right ) {\mathrm e}^{2 \sqrt {-x}} x^{\frac {1}{4}}}{\left (-x \right )^{\frac {1}{4}}}+\frac {c_{2} x^{\frac {1}{4}} \left (2 \sqrt {-x}+1\right ) {\mathrm e}^{-2 \sqrt {-x}}}{8 \left (-x \right )^{\frac {1}{4}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{2 i \sqrt {x}} \sqrt {\frac {\left (1+4 x \right ) \left (2 i \sqrt {x}-1\right )}{1+2 i \sqrt {x}}}+c_{2} {\mathrm e}^{-2 i \sqrt {x}} \sqrt {\frac {\left (1+4 x \right ) \left (1+2 i \sqrt {x}\right )}{2 i \sqrt {x}-1}} \] |
Problem 7787
| ODE |
\[ \boxed {x y^{\prime \prime }+x y^{\prime }-2 y=0} \] |
|
program solution |
\[ y = c_{1} x \left (2+x \right )+c_{2} \left (\frac {\left (-x -1\right ) {\mathrm e}^{-x}}{2}+\frac {\left (2+x \right ) \operatorname {expIntegral}_{1}\left (x \right ) x}{2}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x^{2}+2 x \right )+c_{2} \left (\frac {x^{2} \operatorname {expIntegral}_{1}\left (x \right )}{2}-\frac {{\mathrm e}^{-x} x}{2}+\operatorname {expIntegral}_{1}\left (x \right ) x -\frac {{\mathrm e}^{-x}}{2}\right ) \] |
Problem 7788
| ODE |
\[ \boxed {x \left (x -1\right )^{2} y^{\prime \prime }-2 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} x}{x -1}+\frac {c_{2} \left (-2 \ln \left (x \right ) x +x^{2}-1\right )}{x -1} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \frac {c_{1} x}{x -1}+\frac {c_{2} \left (2 x \ln \left (x \right )-x^{2}+1\right )}{x -1} \] |
Problem 7789
| ODE |
\[ \boxed {y^{\prime \prime }-2 x y^{\prime }+y x^{2}=0} \] |
|
program solution |
\[ y = c_{1} \cos \left (x \right ) {\mathrm e}^{\frac {x^{2}}{2}}+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \sin \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \cos \left (x \right )+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \sin \left (x \right ) \] |
Problem 7790
| ODE |
\[ \boxed {x \left (-x^{2}+2\right ) y^{\prime \prime }-\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right )=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{x} x^{2}+c_{2} \left (1-x \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} \left (x -1\right )+c_{2} {\mathrm e}^{x} x^{2} \] |
Problem 7791
| ODE |
\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-\left (2 x +1\right ) \left (x y^{\prime }-y\right )=0} \] |
|
program solution |
\[ y = \frac {c_{1} \sqrt {x}\, \sqrt {x \left (x +1\right )}}{\sqrt {x +1}}+\frac {c_{2} \sqrt {x}\, \sqrt {x \left (x +1\right )}\, \left (x +\ln \left (x \right )\right )}{\sqrt {x +1}} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} x \left (x +\ln \left (x \right )\right ) \] |
Problem 7792
| ODE |
\[ \boxed {2 \left (2-x \right ) x^{2} y^{\prime \prime }-x \left (-x +4\right ) y^{\prime }+\left (3-x \right ) y=0} \] |
|
program solution |
\[ y = c_{1} \sqrt {x}+2 c_{2} \sqrt {x}\, \sqrt {x -2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {x}+c_{2} \sqrt {x^{2}-2 x} \] |
Problem 7793
| ODE |
\[ \boxed {\left (1-x \right ) x^{2} y^{\prime \prime }+\left (5 x -4\right ) x y^{\prime }+\left (6-9 x \right ) y=0} \] |
|
program solution |
\[ y = c_{1} x^{3}+c_{2} x^{2} \left (\ln \left (x \right ) x +1\right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} x^{3}+c_{2} x^{2} \left (x \ln \left (x \right )+1\right ) \] |
Problem 7794
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (4 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} \ln \left (x \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} \ln \left (x \right ) \] |
Problem 7795
| ODE |
\[ \boxed {y^{\prime \prime }-2 x y^{\prime }+8 y=0} \] |
|
program solution |
\[ y = c_{1} \left (x^{4}-3 x^{2}+\frac {3}{4}\right )+4 c_{2} \left (4 x^{4}-12 x^{2}+3\right ) \left (\int \frac {{\mathrm e}^{x^{2}}}{\left (4 x^{4}-12 x^{2}+3\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (\frac {4}{3} x^{4}-4 x^{2}+1\right )+c_{2} \left (\frac {4}{3} x^{4}-4 x^{2}+1\right ) \left (\int \frac {{\mathrm e}^{x^{2}}}{\left (4 x^{4}-12 x^{2}+3\right )^{2}}d x \right ) \] |
Problem 7796
| ODE |
\[ \boxed {y^{\prime \prime }-2 x y^{\prime }+8 y=0} \] |
|
program solution |
\[ y = c_{1} \left (x^{4}-3 x^{2}+\frac {3}{4}\right )+4 c_{2} \left (4 x^{4}-12 x^{2}+3\right ) \left (\int \frac {{\mathrm e}^{x^{2}}}{\left (4 x^{4}-12 x^{2}+3\right )^{2}}d x \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} \left (\frac {4}{3} x^{4}-4 x^{2}+1\right )+c_{2} \left (\frac {4}{3} x^{4}-4 x^{2}+1\right ) \left (\int \frac {{\mathrm e}^{x^{2}}}{\left (4 x^{4}-12 x^{2}+3\right )^{2}}d x \right ) \] |
Problem 7797
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (5 x^{3}-3 x \right ) \sqrt {x^{2}-1}}{5 \sqrt {x -1}\, \sqrt {x +1}}+\frac {5 c_{2} \left (15 \ln \left (x -1\right ) x^{3}-15 \ln \left (x +1\right ) x^{3}-9 \ln \left (x -1\right ) x +9 \ln \left (x +1\right ) x +30 x^{2}-8\right ) \sqrt {x^{2}-1}}{24 \sqrt {x +1}\, \sqrt {x -1}} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} \left (-\frac {5}{3} x^{3}+x \right )+c_{2} \left (-\frac {5 \ln \left (x +1\right ) x^{3}}{24}+\frac {5 \ln \left (x -1\right ) x^{3}}{24}+\frac {\ln \left (x +1\right ) x}{8}-\frac {\ln \left (x -1\right ) x}{8}+\frac {5 x^{2}}{12}-\frac {1}{9}\right ) \] |
Problem 7798
| ODE |
\[ \boxed {x \left (2+x \right ) y^{\prime \prime }+2 \left (x +1\right ) y^{\prime }-2 y=0} \] |
|
program solution |
\[ y = c_{1} \left (x +1\right )+c_{2} \left (\frac {\left (-x -1\right ) \ln \left (2+x \right )}{2}+1+\frac {\left (x +1\right ) \ln \left (x \right )}{2}\right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} \left (x +1\right )+c_{2} \left (\frac {x \ln \left (x \right )}{2}-\frac {\ln \left (x +2\right ) x}{2}+\frac {\ln \left (x \right )}{2}-\frac {\ln \left (x +2\right )}{2}+1\right ) \] |
Problem 7799
| ODE |
\[ \boxed {x \left (2+x \right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }-4 y=0} \] |
|
program solution |
\[ y = \frac {c_{1} \left (x +1\right ) x^{\frac {3}{4}} \left (2+x \right )^{\frac {3}{4}}}{\left (x \left (2+x \right )\right )^{\frac {1}{4}}}-\frac {2 c_{2} x^{\frac {1}{4}} \left (2+x \right )^{\frac {1}{4}} \left (x^{2}+2 x +\frac {1}{2}\right )}{\left (x \left (2+x \right )\right )^{\frac {1}{4}}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} \left (2 x^{2}+4 x +1\right )+c_{2} \left (x +1\right ) \sqrt {x \left (x +2\right )} \] |
Problem 7800
| ODE |
\[ \boxed {\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{x}-c_{2} x \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = c_{1} x +{\mathrm e}^{x} c_{2} \] |